MakerBot won't slice my model
You have (what looks to be) several orphaned edges in the model that I found by selecting all vertices in Vertex select mode, then switching to Face selection.  I can't say if this is what's causing the error because I don't have MakerBot, but I've removed the edges which serve no purpose.  



EDIT (from your comment): With faces it's a little more complicated than just drawing an edge from point to point, especially when exporting to a different format/program.  The face needs to be planar (existing in a single plane) and no geometry should exist that doesn't support at least one particular face.  In your case, there were multiple 2D edges (in other words, they can't exist in real space... matter always exists in 3D) that I'm guessing Makerbot couldn't resolve because they weren't adding to the box's substance.

For the addition you made to the question, it looks like MakerBot isn't recognizing a face on the outside.  It also looks like it's missing the faces on the rim.  I deleted those faces, added two corner edges for more geometry, and then re-faced the rim and side.

Please try it and let me know, hopefully it will help.  If not, it's difficult to say what's going on: